Affine Harmonic Maps

We introduce a class of maps from an affine flat into a Riemannian manifold that solve an elliptic system defined by the natural second order elliptic operator of the affine structure and the nonlinear Riemann geometry of the target. These maps are called affine harmonic. We show an existence result for affine harmonic maps in a given homotopy class when the target has non positive sectional curvature and some global non triviality condition is met. An example shows that such a condition is necessary. The analytical part is made difficult by the absence of a variational structure underlying affine harmonic maps. We therefore need to combine estimation techniques from geometric analysis and PDE theory with global geometric considerations.